Ion channels are proteins with holes down their middle (some 0.4 - 1 nm in diameter) that control a wide range of biological function and so are studied in thousands of laboratories every day. Ion channels are ideal objects for physical and mathematical investigation because their structure is simple and quite invariant. Ion channels are ideal objects for biological investigation because they control so many cellular functions and are archetypes of an enormous number of proteins, occupying a substantial fraction of the human genome. Ion channels are objects of great clinical importance because they are involved directly in so many diseases. [unreadable] [unreadable] Ion channels can be studied in the physical tradition because ions move by electrodiffusion through a simple invariant structure following well known laws of diffusion in an electric field. Direct simulation of atomic motion in channels is not feasible on the biological time scale of msec if the system is to include well defined concentrations of ions and electrical potentials. The problem is to know what to average and how to average so the function of channels can be understood and controlled. Here we adopt the engineering approach of multiresolution analysis, proposing a mean field theory in the chemical tradition and stochastic theory in the mathematical tradition of stochastic physics. The central issue in both traditions is to extend previous theories of equilibrium systems to allow prediction of the large currents measured in biological channels.